Learning Lab

Definition of an Inverse Function

If \(f^{-1}(x)\) is the inverse funtion of a one-to-one function \(f(x)\) then \(f^{-1}(x)\) is the set of all ordered pairs obtained by interchanging the first and second elements in each ordered pair.

So if \((a,b)\in f\) then \((b,a)\in f^{-1}\) and if \(f(a)=b\) then \(f^{-1}(b)=a\)

The domain of \(f\) is the range of \(f^{-1}\) and the range of \(f\) is the domain of \(f^{-1}\)

For example the function \(f:R\rightarrow R\), defined by \(y=f(x)=\frac{x-1}{2}\) has an inverse function with the rule \(y=2x+1\) .

So \((3,1)\) belongs to \(f\) and \((1,3)\) belongs to \(f^{-1}\) and \((-7,-4)\) belongs to \(f\) and \((-4,-7)\) belongs to \(f^{-1}\) .

Graph of an Inverse Function

The graphs of any one-to-one function, \(f\) , and its inverse, \(f^{-1}\) , are symmetric about the line \(y=x\) .

Finding an Inverse Function for \(y=f(x)\)

To obtain the rule for an inverse function, swap the \(x\) and \(y\) coordinates in \(f\) and rearrange to express \(y\) in terms of \(x\) .